Discrete Math Seminars

Most Tuesdays in K9509 - 12:30pm to 1:20pm

Please join us for this weekly seminar on a wide variety of topics under the umbrella of discrete mathematics. We gratefully acknowledge the Pacific Institute of Mathematical Sciences for their generous funding since 2010.

Watch for upcoming Seminars listed below, offered in SCK 9509 and/or on Zoom. We welcome remote participation from groups at other universities.

All talks will be announced on this page, on the department calendar (found at the bottom of the Math homepage), and on the DM Seminar mailing list. If you have an SFU email address, you can subscribe or unsubscribe to the mailing list by logging in to maillist.sfu.ca and searching for the math-dmsem list. Otherwise, you can ask any discrete math faculty member to manually add you to the list. With the permission of the speaker, we will record the talk video and upload it to the SFU Discrete Math Seminar Youtube Channel

Current organizer: Ladislav Stacho & Alexander Clow

2024 Talks

Spring 2024

Date: Tuesday April 09, 2024 Speaker: Kasra Masoudi, SFU Talk Title: A Strict Inequality on the Energy of Edge Partitioning of Graphs

There is an interesting connection between chemistry and graph theory. This connection is highlighted by the relationship between a energy of a molecule and the eigenvalues of its graph. In the 1970s, a chemist named Ivan Gutman introduced a concept to define the energy of graphs, denoted as E(G). Simply put, Gutman’s definition involves adding up the absolute values of the eigenvalues of the graph’s adjacency matrix A.

There exists a famous inequality on the energy of edge partitioning of Graphs, which states that energy of a graphs G is either less than or equal to the sum of the energies of its edge partitions. In this work, we show that in case of the connectivity of G the inequality is strict by using some features of positive semi-definite matrices and the relationship of the partitions with their adjacency matrices.

Date: Tuesday April 02 , 2024 Speaker: CANCELLED Talk Title: 
 
Date: Tuesday March 26 , 2024 Speaker: Vesna Irsic, University of Ljubljana Talk Title: Domination of subcubic planar graphs with large girth

Since Reed conjectured in 1996 that the domination number of a connected cubic graph of order n is at most \lceil \frac13 n \rceil, the domination number of cubic graphs has been extensively studied. It is now known that the conjecture is false in general, but Henning and Dorbec very recently showed that it holds for graphs with girth at least 9. Zhu and Wu stated an analogous conjecture for connected cubic planar graphs.

In this talk we present a new upper bound for the domination number of subcubic planar graphs: if G is a subcubic planar graph with girth at least 8, then \gamma(G) < n_0 + \frac{3}{4} n_1 + \frac{11}{20} n_2 +  \frac{7}{20} n_3, where n_i denotes the number of vertices in G of degree i, for i \in \{0,1,2,3\}. We also discuss possible improvements of this bound and highlight the main ideas of the proof.

Joint work with Eun-Kyung Cho, Eric Culver and Stephen G. Hartke.

Date: Tuesday March 05 , 2024 Speaker: Amarpreet Rattan, SFU Talk Title: Symmetry in star factorisations (continued)

On Feb 14, J. Campion Loth gave the Discrete Math seminar on "Symmetry of Star Factorisations", where he focused on the combinatorial aspects of these objects and related factorisation problems.  In this talk, we will look at these objects, which I will quickly reintroduce, from the point of view of algebra and the space of symmetric functions.  I will give a brief but gentle introduction to symmetric functions, Jucys-Murphy elements, and their relationship to star and monotone factorisations.

This is joint work with J. Campion Loth.

Date:  Tuesday February 20, 2024   CANCELLED Speaker: Peter Horak, University of Washington Talk Title: Perfect Matchings in Cubic Graphs

In 2008 Alon and Friedland showed that a simple cubic graph G on 2n vertices has at most 6^{n/3} perfect matchings, and this bound is attained by taking the disjoint union of bipartite complete graphs K_{3,3}. In other words, the above theorem says that the complete bipartite graph  K_{3,3} has the highest density of perfect matchings among all cubic graphs; thus the disjoint union of its copies constitutes the extremal graph. However, this result does not provide any insight into the structure of extremal connected cubic graphs.

In this talk it will be presented that for n >= 6, the number of perfect matchings in a simple connected cubic graph on 2n vertices is at most  4f_{n-1}, with f_{n} being the n-th Fibonacci number, and a unique extremal graph will be characterized as well. In addition, it will be shown that the number of perfect matchings in any cubic graph G equals the expected value of a random variable defined on all 2-colorings of edges of G.

In the second part of our talk an improved lower bound on the maximum number of cycles in a cubic graph will be provided.

Tuesday, February 13 , 2024 Sepeaker: Jesse Campion Loth, University of Bristol Talk Title: Symmetry of star factorisations

How many ways are there to write a given permutation as a product of m transpositions (i_1, j_1) (i_2, j_2) \dots (i_m, j_m)?  For example, there are three ways to write (1,2,3) as a product of two transpositions: (1,2)(1,3), (1,3)(2,3) and (2,3)(1,2).  This is an example of a permutation factorisation problem.  There are a broad range of problems of this type, with links to algebraic geometry and graph embeddings.  We will first give an overview of how problems of this type are usually approached using generating functions and character theory.

We then show how a combinatorial approach can be used to give bijections between different classes of permutation factorisation.  In particular, we will present the first fully combinatorial proof of the symmetry of star factorisations, answering a question of Goulden and Jackson [2009].  

This is joint work with Amarpreet Rattan, who will give a seminar on a symmetric function approach to these problems in March.

Date:  Tuesday Feb 06 , 2024 Speaker: Torsten Mutze, University of Warwick Talk Title: Kneser graphs are Hamiltonian

For integers k>=1 and n>=2k+1, the Kneser graph K(n,k) has as vertices all k-element subsets of an n-element ground set, and an edge between any two disjoint sets. It has been conjectured since the 1970s that all Kneser graphs admit a Hamilton cycle, with one notable exception, namely the Petersen graph K(5,2). This problem received considerable attention in the literature, including a recent solution for the sparsest case n=2k+1. The main contribution of our work is to prove the conjecture in full generality. We also extend this Hamiltonicity result to all connected generalized Johnson graphs (except the Petersen graph). The generalized Johnson graph J(n,k,s) has as vertices all k-element subsets of an n-element ground set, and an edge between any two sets whose intersection has size exactly s. Clearly, we have K(n,k)=J(n,k,0), i.e., generalized Johnson graphs include Kneser graphs as a special case. Our results imply that all known families of vertex-transitive graphs defined by intersecting set systems have a Hamilton cycle, which settles an interesting special case of Lovász’ conjecture on Hamilton cycles in vertex-transitive graphs from 1970. Our main technical innovation is to study cycles in Kneser graphs by a kinetic system of multiple gliders that move at different speeds and that interact over time, reminiscent of the gliders in Conway’s Game of Life, and to analyze this system combinatorially and via linear algebra.

This is joint work with my students Arturo Merino (TU Berlin) and Namrata (Warwick).

 

Fall 2023

Date: Tuesday December 5, 2023 Speaker: Omer Angel, UBC Talk Title: Non-monotonicity for all-in poker tournaments

We consider a simplified model for a poker tournament taking into account only hands where two or more players make the maximal bet of their entire stack.  While the probability of any player being the overall winner is a simple function of the amounts they currently have, the probability of ending up in kth place for any other k is a surprisingly complex function, which we study. 

Joint work with Mark Holmes.

Date:  Tuesday November 28, 2023 Speaker: Yanwen Luo, SFU Talk Title: Spaces of Geodesic Triangulations of Surfaces

In 1962, Tutte proposed a simple method to produce a straight-line embedding of a planar graph in the plane, known as Tutte's spring theorem. This construction provides not only one embedding of a planar graph, but infinite many distinct embeddings of the given graph. This observation leads to a surprisingly simple proof of a classical theorem proved by Bloch, Connelly, and Henderson in 1984 stating that the space of geodesic triangulations of a convex polygon is contractible. In this talk, we will introduce spaces of geodesic triangulations of surfaces, review Tutte's spring theorem, and present this short proof. We will briefly report the recent progress in identifying the homotopy types of spaces of geodesic triangulations of more complicated surfaces. 

This is a joint work with Tianqi Wu and Xiaoping Zhu.

Date: Tuesday November 7, 2023 Speaker: Speaker: Yifan Jing, University of Oxford Talk Title: Measure doubling for sets in locally compact groups

A central problem in additive combinatorics is to obtain a good lower bound for the size of the product set AB of two subsets A, B of a group G. In this talk, we will be focusing on the setting when G is a locally compact group, where G has a natural notion of size: the Haar measure. I will talk about some recent developments and their connections to analysis and geometry, including the resolution of the Inverse Kemperman problem, a non-abelian Brunn-Minkowski inequality, and the confirmation of a conjecture of Breuillard and Green. 

The talk is based on joint work with Chieu-Minh Tran and Ruixiang Zhang.

Date:  Tuesday October 31, 2023 Speaker: Shivaramakrishna Pragada, SFU Talk Title: Subdivision and adjacency spectra of graphs

In this talk, we investigate the asymptotic nature of graph spectra when some edges of a graph are subdivided sufficiently many times. We show that the eigenvalues of the sequences of graphs obtained by subdividing edges are Cauchy. As an application of the main result, we construct a bounded degree graph sequence with (approximate) high second eigenvalue multiplicity. 

This is a joint work with Hitesh Kumar, Bojan Mohar and Hanmeng Zhan.

Date:  Tuesday October 24, 2023 Speaker: Thomas Pender, SFU Talk Title: Balanced Weighing Matrices and Association Schemes

A weighing matrix is a square (-1,0,+1)-matrix with pairwise orthogonal rows such that each row has a constant number of non-zero entries. If upon taking the absolute values of all the entries of a weighing matrix, one obtains an incidence matrix for a symmetric balanced incomplete block design, we say that the weighing matrix is balanced.

In this talk, we construct a new infinite family of balanced weighing matrices. We then show a novel characterization of balanced weighing matrices using association schemes. Finally, we explore analogous relations between more general classes of objects: balanced generalized weighing matrices, quasi-balanced weighing matrices.

This is joint work with Dr. Hadi Kharaghani (University of Lethbridge) and Dr. Sho Suda (National Defense Academy of Japan).

Date:  Tuesday October 3, 2023 Speaker: Shizhe Liang, Brandeis University Talk Title: Grand-Schnyder woods and applications to drawing

In 1990, Schnyder showed that every plane triangulation admits a special partition of its inner edges into 3 trees, which are known as the Schnyder woods. Using this structure, Schnyder designed a straight-line grid drawing algorithm for triangulations. Since then, other structures and algorithms sharing the same flavor have been found by various authors.

In this talk I will present a new construction, named grand-Schnyder woods, that simultaneously generalizes several of these structures (notably the original Schnyder woods, the regular edge labelings of Kant and He, and the d-Schnyder structures of Bernardi and Fusy). Precisely, a grand-Schnyder wood of parameter d is defined on a planar graph which has faces of degree at most d and non-facial cycles of length at least d. It is a set of d spanning forests crossing each other in a specific way. 

Grand-Schnyder woods of parameter d=4 inspires a general framework of drawing algorithms, which unifies several classical algorithms due to Kant and He, Fusy, Barriere & Huemer, and Bernardi & Fusy.

This is a joint work with Olivier Bernardi and Eric Fusy.

Date:  Tuesday September 26, 2023 Speaker: Alexander Clow, SFU Talk Title: Injective and Oriented Colouring Graphs of Bounded Euler Genus

In this talk we consider the injective chromatic index and the oriented chromatic number of graphs with bounded Euler genus. In particular, we present our proofs of a tight (up to the choice of  little o(1)) bound on the injective chromatic index in genus and that the oriented chromatic number is bounded polynomially in genus.  The latter being a major improvement over the previous best upper bound which is exponential in genus. We conclude the talk by discussing directions for future study.

Joint work with Peter Bradshaw (University of Illinois Urbana Champaign), and Jingwei Xu (University of Illinois Urbana Champaign).

Date:  Tuesday September 19, 2023 Speaker: Maxwell Levit, SFU Talk Title: 4-cycle-free covers of Cartesian products of cycles
In order to classify certain small generalized hexagons, Cohen and Tits proved that the d-dimensional hypercube has a unique 2-fold cover with no 4-cycles. I will discuss some reasons I find these covers interesting, including a close connection to Huang's recent proof of the Sensitivity conjecture from computer science.  Then I will give a group-theoretic description of these covers which generalizes nicely to provide 4-cycle-free covers of Cartesian products of p-cycles.